Regularity of optimal transport maps

on multiple products of spheres ∗

Alessio Figalli†, Young-Heon Kim‡ and Robert J. McCann§

October 11, 2011

Abstract

This article addresses regularity of optimal transport maps for cost=“squared distance”on Riemannian manifolds that are products of arbitrarily many round spheres with arbi-trary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved[KM2]. Under boundedness and non-vanishing assumptions on the transfered source andtarget densities we show that optimal maps stay away from the cut-locus (where the costexhibits singularity), and obtain injectivity and continuity of optimal maps. This allowsto be used to with the results of Liu, Trudinger and Wang [LTW] to conclude higher reg-ularity (C1,α/C∞) of optimal maps for smoother (Cα/C∞) densities. These are the firstglobal regularity results which we are aware of concerning optimal maps on Riemannianmanifolds which possess some vanishing sectional curvatures, beside the totally flat case ofRn [Ca3] and its quotients [Co]. Moreover, such product manifolds have potential relevancein statistics (see [S]) and in statistical mechanics (where the state of a system consisting ofmany spins is classically modeled by a point in the phase space obtained by taking manyproducts of spheres). For the proof we apply and extend the method developed in [FKM1],where we showed injectivity and continuity of optimal maps on domains in Rn for smoothnon-negatively cross-curved cost. The major obstacle in the present paper is to deal withthe product structure of the cut-locus and the presence of flat directions.

Optimal transport, functional inequalities and Riemannian geometry

Contents

1 Introduction 2

2 Notation and assumptions 5∗The authors are grateful to the Institute for Pure and Applied Mathematics at UCLA and the Institute for

Advanced Study in Princeton for their generous hospitality during various stages of this work. AF is partiallysupported by NSF grant DMS-0969962. RJM is supported in part by NSERC grants 217006-08 and NSF grantDMS-0354729. YHK is supported partly by NSF grant DMS-0635607 through the membership at Institute forAdvanced Study at Princeton NJ, and also in part by NSERC grant 371642-09. Any opinions, findings andconclusions or recommendations expressed in this material are those of authors and do not reflect the views ofeither the Natural Sciences and Engineering Research Council of Canada (NSERC) or the United States NationalScience Foundation (NSF). c©2010 by the authors.

†Department of Mathematics, The University of Texas at Austin, Austin TX, USA. figalli@math.utexas.edu‡Department of Mathematics, University of British Columbia, Vancouver BC, Canada. yhkim@math.ubc.ca§Department of Mathematics, University of Toronto, Toronto ON, Canada. mccann@math.toronto.edu

1

3 Preliminary results 93.1 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Coordinate change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Relation between cotangent vectors in two different coordinates . . . . . . 113.2.2 An estimate on the first derivatives of c . . . . . . . . . . . . . . . . . . . 12

4 An Alexandrov estimate: lower bound 12

5 Main result: Stay-away property on multiple products of spheres 15

6 Proof of Theorem 5.1 (Stay-away from cut-locus) 186.1 Cut-exposed points of contact sets . . . . . . . . . . . . . . . . . . . . . . . . . . 186.2 Analysis near the cut-exposed point . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 An Alexandrov type estimate near the cut-exposed point . . . . . . . . . . . . . . 236.4 Proof of Theorem 6.4 (Alexandrov upper bound near cut-exposed point): analysis

in the cut-locus component M ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.5 Proof of Theorem 6.4 (Alexandrov upper bound near cut-exposed point): analysis

in the regular component M ′′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.6 Proof of Theorem 6.4 (Alexandrov upper bound near cut-exposed point): final

argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Aubry Mather Theory and Optimal Transpor Nonlinear Hyperbolic PDEs, Dispersive and

Transport Equation: Analysis and Control

1 Introduction

ERC-Summer School on ”Calculus of Variations, Continuum Mechanics and Geometric Inequal-ities” Let M and M be n-dimensional complete Riemannian manifolds, and let µ = ρ volM andν = ρ volM be two probability measures whose densities ρ and ρ are bounded away from zeroand infinity. Given a cost function c : M × M → R, the optimal transport problem with costc(x, y) consist in finding a transport map T : M → M which sends µ onto ν and minimizes thetransportation cost ∫

Mc(x, T (x)) dµ(x).

As shown by McCann [M] extending the result of Brenier [Br] on Rn, if M = M and c =dist2 /2 then the optimal transport map (or simply optimal map) exists and is unique. Moregenerally, the same result holds if the cost is semiconcave and satisfies the twist condition inAssumption 2.1, see for instance [V, Chapter 10].

The optimal map T is uniquely characterized by the relation T (x) ∈ ∂cφ(x), where φ is ac-convex function (called potential) and ∂cφ denotes its c-subdifferential (see Section 2 for thedefinitions). Furthermore, the fact that ρ and ρ are bounded away from zero and infinity ensuresthe existence of a constant λ > 0 such that the following Monge-Ampere type equation holds:

λ |Ω| ≤ |∂cφ(Ω)| ≤ 1λ|Ω| ∀Ω ⊂ M Borel,

where ∂cφ(Ω) = ∪x∈Ω∂cφ(x). (See for instance [FKM1, Lemma 3.1].)The aim of this paper is to investigate the regularity issue of optimal maps when M = M are

multiple product of spheres, i.e., M = M = Sn1r1

× . . .× Snkrk

, and c(x, y) = f(dist(x, y)) for some

2

function f , including the case f(t) = t2/2 of distance squared cost. For k = 1 and f(t) = t2/2,smoothness of optimal maps has been proved by Loeper [L2]. However, if k > 1 the structureof the cut-locus (the singular set of the cost function) becomes more complicated, and due tothe product structure, the manifold has both flat and positively curved directions, thus makingthe regularity issue much more delicate. Especially, the powerful Holder regularity estimate ofLoeper [L1] (see also [Li]) as well as the a priori estimates of Ma, Trudinger and Wang [MTW],which are successfully applied to positively curved manifolds as in [L2, KM2, LV, FR, DG, FRV],are not available any more in our setting. Our main results (Theorem 5.1 and Corollary 5.3)give the first global regularity results which we are aware of concerning optimal maps on non-flatRiemannian manifolds which allow vanishing sectional curvature. For completely flat manifolds(with c = dist2/2) the regularity of optimal maps is known as it reduces to the regularity theoryof the classical Monge-Ampere equation [D1, Ca1, Ca2, Ca3, U, Ca4, Co, D2, G].

To describe our result more precisely, first recall that in [MTW] Ma, Trudinger and Wangdiscovered condition (A3) on the cost function, whose weaker variant (A3w) [TW] turned outto be both necessary [L1] and sufficient [TW] for regularity when the solution φ is known to bestrictly c-convex and the cost function is smooth. When M = M = Sn

r , the particular structureof the cut-locus (for every point x, its cut-locus Cut(x) consists of its antipodal point) allowedDelanoe and Loeper [DL] to deduce that optimal maps stay away from the cut-locus, namely,∂cφ(x)∩Cut(x) = ∅ for all x ∈ M ; see [L2, DG, KM1, KM1a] for alternate approaches. Loeper[L2] combined this observation with the fact that c = dist2 /2 satisfies (A3) to show regularityof optimal maps; for a simpler approach to continuity, see [KM1, KM1a]. His result has beenextended to variety of positively curved manifolds including the complex projective space [KM2]and perturbation of the real projective space [LV] and of the sphere [FR, DG, FRV], all of where(A3) holds thus the strong Holder regularity estimate of [L1] as well as the a priori estimateof [MTW] applies. Note that (A3) (resp. (A3w)) forces the sectional curvature to be positive(resp. nonnegative) [L1], though the converse does not hold [K].

On multiple products of spheres, taking c = dist2 /2 leads to two main issues: first, only adegenerate strengthening of the weak Ma-Trudinger-Wang condition holds (the so-called non-negative cross-curvature condition in [KM1, KM2]), which although stronger than (A3w) is notas useful as (A3) for proving regularity due to lack of powerful estimates; (neither non-negativecross-curvature nor (A3) implies the other, though either one separately implies (A3w)). More-over, the cut-locus now has a non-trivial structure, which makes it much more difficult to un-derstand whether the stay-away property holds. In [FKM1] we showed strict c-convexity andC1 regularity of φ, or equivalently, injectivity and continuity of T , when the cost is smoothand non-negative cross-curvature holds. Hence the only question left is whether ∂cφ avoids thecut-locus or not.

In this paper we answer this question positively: by taking advantage of the fact that thecut-locus is given by the union of certain sub-products of spheres we prove in Theorem 5.1 thestay-away property that ∂cφ(x) ∩ Cut(x) = ∅ for all x ∈ M . By compactness, these two setsare separated by a uniform distance that is dependent on λ, but independent of the particularchoice of φ and x; see Corollary 5.2. Once stay-away property is shown, one can localizethe argument of [FKM1] to obtain injectivity and continuity of the optimal map; then higherregularity (C2,α/C∞) of φ, thus C1,α/C∞-regularity of T , follows from [LTW] when the densitiesare smooth (Cα/C∞); see Corollary 5.3.

The multiple products of spheres is a model case for more general manifolds on which thecost c satisfies the necessary conditions [L1, FRV] for regularity of optimal transport maps. Themethod we develop in this paper demonstrates one approach to handling complex singularities

3

of the cost, especially the stay-away property of optimal maps. Moreover, a general Alexandrovtype estimate (Lemma 4.1) is obtained which has applications beyond the products of spheres.

Our regularity result has potential relevance to statistics and statistical mechanics. Forinstance, recently T. Sei applied optimal transport theory for c = dist2 to directional statisticson the sphere. In his main result [S, Theorem 1], he needed the optimal map not to touchthe cut-locus. Now, our stay-away property on multiple products of spheres M (Theorem 5.1)states that all optimal maps, obtained by transporting densities bounded away from zero andinfinity onto each other, satisfy this assumption. Hence, this provides a large family of c-convexpotentials that could be used to create log-concave likelihood functions as in [S, Subsection3.2], extending his theory to multiple products of spheres. Namely, as a direct consequenceof [KM2, FKM2, S], on multiple products of spheres a convex combination φ =

∑ki=1 siφi,

si ≥ 0,∑

si = 1, of c-convex functions φi is again c-convex, thus a crucial requirement in Sei’stheory is satisfied. If each φi is the c-potential of an optimal map between densities boundedaway from zero and infinity, by Theorem 5.1 one sees ∂cφi stay away from the cut-locus. Onethen can show that ∂cφ also avoids the cut-locus, thus applying [S, Theorem 1] one obtains thelog-concave Jacobian inequality for this convex combination. To see this, for example, observethat in the product of spheres the “domain of exponential map” is convex1, and ∂cφ satisfies∂cφ(x) = expx ∂φ(x) = expx

[∑i si∂φi(x)

]for x ∈ M (see Lemma 2.7). Since each ∂cφi stays

away from the cut-locus, ∂φi(x) belongs to the domain of exponential map, so does ∂φ(x),showing ∂cφ(x) ∩ Cut(x) = ∅.

Concerning statistical mechanics, let us recall that the state of a spin system is classicallymodeled as a point in the phase space M obtained by taking many products of spheres. In suchcontexts, optimal transport may provide a useful change of variables. More precisely, if µ and νare two smooth densities and T denotes the optimal transport map from µ to ν, then∫

G(y) dν(y) =∫

G(T (x)) dµ(x),

for all bounded measurable functions G : M → R. Then, if µ is a “nice” measure forwhich many statistical quantities are easily computable, one may hope to exploit some quali-tative/quantitative properties of T in order to estimate the integral

∫G(y) dν(y) by studying∫

G(T (x)) dµ(x). We expect that regularity of optimal maps may play a crucial role in thisdirection. For instance, in Euclidean spaces this is already the case, as Caffarelli [Ca5] usedregularity of optimal maps to show that suitable monotonicity and log-concavity properties ofthe densities imply monotonicity and contraction properties for the optimal map, from whichcorrelation and momentum inequalities may be deduced.

Organization of the paper: Section 2 sets up the notation and assumptions used through-out the paper. In Section 3, a few useful preliminary results regarding convex sets and c-convexfunctions are listed. Section 4 is devoted to an Alexandrov type inequality which is one of themain tools in the proof of our main theorem. Until Section 4, we present the theory underrather general assumptions. However, from Section 5 we restrict to the multiple products ofspheres. In Section 5 we state our main result about the stay-away property of optimal maps,and give a sketch of the proof. Moreover we explain how one can deduce regularity of optimalmaps combining this theorem with the results in [FKM1] and [LTW]. Finally, the details of the

1Here and in the sequel we use “domain of exponential map” as a synonymous of “injectivity domain”, seeSection 2.

4

proof of the stay-away property are given in Section 6.

Acknowledgement: The authors are pleased to thank Neil Trudinger, Tom Spencer, andCedric Villani for useful discussions.

2 Notation and assumptions

In this section and the next we recall notation and results which will be useful in the sequel.Many of these results originated in or were inspired by the work of Ma, Trudinger, Wang[MTW] and Loeper [L1]. Though the present paper mainly concerns the Riemannian distancesquared cost c = dist2 /2 on the product of round spheres, we will present our work in a rathergeneral framework. It requires only a small additional effort and may prove useful for furtherdevelopment and applications of the theory.

Let M , M be n-dimensional complete Riemannian manifolds, and let c(x, x) denote a costfunction c : M × M → R. We will assume through the whole paper that c is semiconcavein both variables, i.e., in coordinate charts it can be written as the sum of a concave and asmooth function. Let us remark that since dist2(x, y) is semiconcave on M ×M (see for example[FF, Appendix B]), the above assumption is satisfied for instance by any cost function of theform f(dist(x, y)) on M × M , with f : R → R smooth, even, and strongly convex (meaningf(d) = f(−d) and f ′′(d) > 0 for all d ≥ 0). Here and in the sequel we use smooth as asynonymous of C∞ (though C4 would be enough for all our purposes).

As for x and M , we use the “bar” notation to specify the second variable of the cost function.Also as a notation we use c(x, x) := c(x, x). We denote by Dx and Dx the differentials withrespect to the x and x variable respectively. (For instance, DxDxc(x0, x0) denotes the mixedpartial derivative of c at (x0, x0).) Let c-Cut(x) denote the c-cut-locus of x ∈ M , that is

c-Cut(x) := x ∈ M | c is not smooth in a neighborhood of (x, x),

and let M(x) denote the c-injectivity locus M \ c-Cut(x). Define c-Cut(x), M(x) similarly.These sets are open.

Assumption 2.1 (twist). For each (x, x) ∈ M × M , the maps −Dxc(x, ·) : M(x) → T ∗xM and

−Dxc(·, x) : M(x) → T ∗xM are smooth embeddings (thus injective).

We remark that the above hypothesis from Levin [L] is equivalent to condition (A1) in[MTW, L1, KM1], which together with the semiconcavity of the cost ensures existence anduniqueness of optimal maps when the source measure is absolutely continuous with respect tothe volume measure (see for instance [L, FF, F] or [V, Chapter 10]).

The domain of the c-exponential M∗(x) in T ∗xM is defined as the image of M(x) under the

map −Dxc(x, ·), i.e.,

M∗(x) := −Dxc(M(x), x) ⊂ T ∗xM.

Define M∗(x) similarly.As in [MTW, L1], we define the c-exponential maps c-Expx : M∗(x) ⊂ T ∗

xM → M andc-Expx : M∗(x) ⊂ T ∗

xM → M as the inverse maps of −Dxc(x, ·) and −Dxc(·, x) respectively,i.e.,

p = −Dxc(x, c-Expxp) for p ∈ M∗(x), p = −Dxc(c-Expxp, x) for p ∈ M∗(x).

5

Given a set X, we denote by cl(X) its closure. Define the subdifferential of a semiconvexfunciton α : M → R at x ∈ M by

∂α(x) := p ∈ T ∗xM | α(expx v) − α(x) ≥ 〈p, v〉 + o(|v|x) as v → 0 in TxM

(This is non-empty at every point.) Here 〈·, ·〉 denotes the paring of covectors and vectors.

Assumption 2.2. For each (x, x) ∈ M×M the map c-Expx (resp. c-Expx) extends to a smoothmap from cl(M∗(x)) (resp. cl(M∗(x))) onto M (resp. M). If we abuse notation to use c-Expx,c-Expx to denote these extensions, then they satisfy

c-Expxp = c-Expx

(∂x

(− c(x, c-Expxp)

)),∀p ∈ cl(M∗(x));

c-Expxp = c-Expx

(∂x

(− c(c-Expxp, x)

)),∀p ∈ cl(M∗(x)).

Here, ∂x, ∂x denote the subdifferentials with respect to the variables x, x, respectively.

Note that the above assumptions hold for instance when M = M and c = dist2 /2 (so thatc-Expx coincides with the Riemannian exponential map expx). However, the following threeassumptions are much more restrictive, and not true for c = dist2 /2 in general [MTW, L1,KM1, LV]. They are all crucial in this paper.

Assumption 2.3 (convexity of domains of c-exponentials). For each (x, x) ∈ M × M thedomains M∗(x), M∗(x) are convex.

As shown in [FRV], the above assumption is necessary for continuity of optimal transportmaps when the cost function is given by the squared distance.

A c-segment x(t)0≤t≤1 with respect to x is the c-exponential image of a line segment incl(M∗(x)), i.e.,

x(t) := c-Expx((1 − t)p0 + tp1), for some p0, p1 ∈ T ∗xM .

Define similarly a c-segment x(t)0≤t≤1 with respect to x. The notions of c- and c-segments,due to Ma, Trudinger and Wang, induce a natural extension of the notion of convexity on setsin M , M called c-convexity in [MTW]. Let U ⊂ M , x ∈ M . The set U is said to be c-convexwith respect to x if any two points in U are connected by a c-segment with respect to x entirelycontained inside U . Similarly we define c-convex sets in M . It is helpful to notice that c, c-convex sets (with respect to x, x, respectively) are images of convex sets under c-Expx, c-Expx,respectively.

Regarding c, c-segments, here comes a key assumption in this paper:

Assumption 2.4 (convex DASM). For every (x, x) ∈ M × M , let x(t)0≤t≤1, x(t)0≤t≤1

be c, c-segments with respect to x, x, respectively. Define the functions

mt(·) := −c(·, x(t)) + c(x, x(t)), mt(·) := −c(x(t), ·) + c(x(t), ·), 0 ≤ t ≤ 1.

Then

mt ≤ (1 − t)m0 + t m1, mt ≤ (1 − t)m0 + t m1 0 ≤ t ≤ 1. (2.1)

6

When, instead of (2.1), only mt ≤ max[m0,m1] and mt ≤ max[m0, m1] are required, thisproperty played a key role in the work of Loeper [L1]. In [KM1] we called it Loeper’s maxi-mum principle (DASM), the acronym (DASM) standing for “Double Mountain Above SlidingMountain”, a mnemonic which describes how the graphs of the functions mt, mt behave as tis varied. For convenience we use this acronym in various places in the present paper. Thestronger property (convex DASM) was proved in [KM2] to be a consequence of the so-callednonnegative cross-curvature condition on the cost c.

We will also need a strict version of Loeper’s maximum principle (DASM):

Assumption 2.5 (DASM+). With the same notation as in Assumption 2.4,

mt(y) ≤ max[m0(y),m1(y)] ∀ y ∈ M, mt(y) ≤ max[m0(y), m1(y)] ∀ y ∈ M.

Moreover, when the c-(resp. c-)segument in the definition of mt (resp. mt) is nonconstant, theequality holds if and only if y = x (resp. y = x).

Assumptions 2.4 and 2.5 correspond to a “global” version of the non-negative cross curvatureassumption and of the (A3) condition of the cost function c, respectively: see [KM1] and[MTW] for the definition of nonnegative cross curvature and (A3), respectively. Although theequivalence between (convex DASM) and non-negative cross curvature (resp. (DASM+)and (A3)) is not known in general, it holds true for the squared distance cost function on aRiemannian manifold, as shown in [FV, FRV]. Moreover, Loeper’s maximum principle (DASM)is a necessary condition for regularity: this is originally shown [L1] on domains in Rn and laterextended to the manifold case [FRV].

Given two functions φ : M → R and φ : M → R, we say that they are c-convex and dualwith respect to each other if

φ(x) = supx∈M

−c(x, x) − φ(x), (2.2)

φ(x) = supx∈M

−c(x, x) − φ(x) = supx∈M

−c(x, x) − φ(x).

Since by assumption c is semiconcave, both functions above are semiconvex (see for instance [FF,Appendix A]). This implies in particular that their subdifferentials, ∂φ(x), ∂φ(x) are non-emptyat every point.

We define the c-subdifferential ∂cφ at a point x as follows:

∂cφ(x) := x ∈ M | φ(y) − φ(x) ≥ −c(y, x) + c(x, x), ∀ y ∈ M. (2.3)

Analogously, we define ∂ cφ at every point x. (Recall that c denotes the function defined asc(x, x) := c(x, x).) The following well-known reciprocity holds:

Lemma 2.6 (Reciprocity). For c-convex functions φ, φ dual to each other as in (2.2),

x ∈ ∂cφ(x) ⇐⇒ φ(x) + φ(x) = −c(x, x) ⇐⇒ x ∈ ∂ cφ(x). (2.4)

Proof. Suppose x ∈ ∂cφ(x). Then, by rearranging the inequality in (2.3) we get

−φ(x) ≥ c(x, x) + supy∈M

−c(y, x) − φ(y),

7

and the supremum on the right hand side is exactly φ(x). On the other hand, from the definitionof φ and φ we have

φ(y) + c(x, y) ≥ −φ(x) ∀ y ∈ M,

so that combining these two inequalities leads to φ(x) + φ(x) = −c(x, x), and x ∈ ∂ cφ(x). Theopposite implication follows by symmetry.

Loeper [L1] deduced the following fundamental relation to be a consequence of his maximumprinciple (DASM).

Lemma 2.7 (Loeper’s maximum principle (DASM)). Let Assumptions 2.1, 2.2 and 2.3hold. Suppose Loeper’s maximum principle (DASM) holds. Let φ, φ be c-convex functions dualto each other as in (2.2). Then for all x ∈ M, x ∈ M ,

c-Expx(∂φ(x)) = ∂cφ(x), c-Expx(∂φ(x)) = ∂ cφ(x).

Proof. The inclusions c-Expx(∂φ(x)) ⊂ ∂cφ(x), c-Expx(∂φ(x)) ⊂ ∂ cφ(x) follow from the convex-ity of ∂φ(x) and the definition of Loeper’s maximum principle (DASM). The other inclusionshold in general without Loeper’s maximum principle. Details can be found in [L1].

In the following, we refer the conclusion of this lemma also as Loeper’s maximum principle(DASM).

For a set Ω ⊂ M , the image ∂cφ(Ω) is defined as

∂cφ(Ω) :=∪x∈Ω

∂cφ(x).

For a c-convex function φ and an open set U ∈ M with x0 ∈ U , we define the set [∂cφ(U)]x0 ⊂ Mas

[∂cφ(U)]x0 := x ∈ M | φ(x) − φ(x0) ≥ −c(x, x) + c(x0, x) for all x ∈ ∂U.

Trivially, ∂cφ(x0) ⊂ [∂cφ(U)]x0 . This definition is justified by the following lemma, which is alsovery useful in later discussions.

Lemma 2.8. Let Assumptions 2.1, 2.2 and 2.3 hold. Suppose Loeper’s maximum principle(DASM) holds. Let φ be a c-convex function on M . Let U ⊂ M be an open set, and letx0 ∈ U . Then

(1) [∂cφ(U)]x0 is c-convex with respect to x0;

(2) [∂cφ(U)]x0 ⊂ ∂cφ(U);

(3) If U → x0, then both ∂cφ(U), [∂cφ(U)]x0 → ∂cφ(x0).

Proof. Assertion (1) follows directly from the definitions of Loeper’s maximum principle (DASM)and of the set [∂cφ(U)]x0 .

To prove Assertion (2), fix x ∈ [∂cφ(U)]x0 , and move first the graph of the function −c(·, x)down so that it lies below φ inside U , and then lift it up until it touches the graph of φ insidecl(U). Thanks to the assumption x ∈ [∂cφ(U)]x0 there exists at least one touching point x′

which belongs to U (indeed, if there is a touching point on ∂U , then x0 is another touchingpoint), and Lemma 2.7 ensures that x ∈ ∂cφ(x′).

For (3), the convergence ∂cφ(U) → ∂cφ(x0) follows by continuity, and [∂cφ(U)]x0 → ∂cφ(x0)comes then from (2).

8

For x ∈ M , let S(x) be the contact set

S(x) := x ∈ M | x ∈ ∂cφ(x) = ∂ cφ(x).

(The last identity follows from reciprocity, see Lemma 2.6.) For any x0 ∈ S(x) one can write

S(x) = x ∈ M | φ(x) − φ(x0) = −c(x, x) + c(x0, x).

A set Z in M is called a c-section of φ with respect to x if there is λx ∈ R such that

Z := z ∈ M | φ(z) ≤ −c(z, x) + λx .

The following simple observation is very useful for studying regularity of c-convex functions.It was originally made (implicitly) in [FKM1] and independently by Liu [Li].

Lemma 2.9 (c-convex c-sections). Let Assumptions 2.1, 2.2 and 2.3 hold. Suppose Loeper’smaximum principle (DASM) holds. Let φ be a c-convex function on M , and fix x ∈ M . Everyc-section Z of φ with respect to x is c-convex with respect to x.

Proof. This follows from the definition of c-convex functions and Loeper’s maximum principle(DASM).

Given Borel sets V ⊂ M and V ⊂ M , we denote by |V | and |V | their volume (computedwith respect to the given Riemannian metric on M and M , respectively). The following is ourlast assumption. As we already remarked in the introduction, it is satisfied whenever φ is thepotential associated to an optimal transport map and the densities are both bounded away fromzero and infinity.

Assumption 2.10 (bounds on c-Monge-Ampere measure of φ). There exists λ > 0 suchthat

λ|Ω| ≤ |∂cφ(Ω)| ≤ 1λ|Ω| for all Borel set Ω ⊂ M.

We sometimes abbreviate this condition on φ simply by writing |∂cφ| ∈ [λ, 1λ ].

3 Preliminary results

In this section, we list some preliminary results we require later. The first subsection deals withgeneral convex sets and the second subsection considers the properties of the cost function undersuitable assumptions.

3.1 Convex sets

We first list two properties of convex sets that will be useful later.

Lemma 3.1 (John’s lemma). For a compact convex set S ⊂ Rn, there exists an affine trans-formation L : Rn → Rn such that B1 ⊂ L−1(S) ⊂ Bn. Here, B1 and Bn denote the ball ofradius 1 and n, respectively, centered at 0.

Proof. See [J].

9

Lemma 3.2. Let S be a convex set in Rn = Rn′ × Rn′′, and denote by π′, π′′ the canonical

projections onto Rn′and Rn′′

, respectively. Let S′ be a slice orthogonal to the second component,that is

S′ = (π′′)−1(x′′) ∩ S for some x′′ ∈ π′′(S).

Then there exists a constant C(n), depending only on n = n′ + n′′, such that

C(n) |S| ≥ H n′(S′)H n′′

(π′′(S)),

where H d denotes the d-dimensional Hausdorff measure.

Proof. See [FKM1, Lemma 6.11].

The following lemma is important in the last step (Section 6.6) of the proof of the maintheorem.

Lemma 3.3. Let X = X1 × . . . × Xk, with X i = Rni, i = 1, . . . , k, and write a point x ∈ Xas x = (x1, . . . , xk), xi ∈ Xi. For each i = 1, . . . , k, let U i be a subset of Xi, and let si =(s1

i , . . . , ski ) ∈ X with si

i ∈ U i. Define Si ⊂ X as

Si := s1i × . . . × si−1

i × Ui × si+1i × . . . × sk

i ,

and consider the convex hull co(S1, . . . , Sk) of the sets S1, . . . , Sk. Then there exists a constantC(n, k), depending only on n := n1 + . . . + nk and k, such that

C(n, k) | co(S1, . . . , Sk)| ≥ Πki=1H

ni(Si).

Proof. First consider the barycenter b of the set s1, . . . , sk, that is

b :=1k(s1 + . . . + sk).

We will construct sets Sbi each of which contains b and has Hausdorff measure comparable with

Si. In addition, these sets are mutually orthogonal. We will finish the proof by considering thevolume of the convex hull of these sets Sb

1, . . . , Sbk.

For each i, let bi be the barycenter of the set s1, . . . , sk \ si, i.e.,

bi :=1

k − 1(s1 + . . . + si−1 + si+1 + . . . + sk).

Consider the cone co(bi, Si) ⊂ co(S1, . . . , Sk) and let Sbi be the intersection

Sbi := co(bi, Si) ∩ x ∈ X | xj = bj for j 6= i.

Note that b ∈ Sbi and these sets Sb

1, . . . , Sbk are mutually orthogonal, in the sense that, for each

x ∈ Sbi and y ∈ Sb

j with i 6= j, it holds (x − b) · (y − b) = 0. Now, consider the convex hullco(Sb

1, . . . , Sbk) ⊂ co(S1, . . . , Sk). The previous orthogonality implies

C(n) | co(Sb1, . . . , S

bk)| ≥ Πk

i=1Hni(Sb

i ).

for some constant C(n) depending only on n1+ . . .+nk. (This inequality is obtained for instanceby iteratively applying Lemma 3.2.) To conclude the proof simply observe that b = 1

ksi + k−1k bi,

and soH ni(Sb

i ) ≥1

kniH ni(Si).

10

3.2 Coordinate change

In this subsection we briefly recall the coordinate change introduced in [FKM1, Section 4] thattransforms c-convex functions into convex functions under the condition (convex DASM),referring to [FKM1, Section 4] for more details. Throughout this subsection we let Assump-tions 2.1, 2.2 and 2.3 hold.

Let y0 be an arbitrary point in M . Then the map x ∈ M(y0) 7→ q ∈ T ∗y0

M given byq(x) = −Dxc(x, y0) is an embedding thanks to Assumption 2.1. Recall that M∗(y0) ⊂ T ∗

y0M

denotes the image of this map, that this map is by definition the inverse c-exponential map(c-Expy0

)−1, and the c-exponential map is a diffeomorphism up to the boundary of M∗(y0) (seeAssumption 2.2). Denote

c(q, x) := c(x(q), x) − c(x(q), y0).

Then the c-convex function φ is transformed to a c-convex function ϕ defined as

ϕ(q) := φ(x(q)) + c(x(q), y0).

If Loeper’s maximum principle (DASM) holds, then Lemma 2.9 shows that c-sections of ϕ areconvex. This property was observed independently by Liu [Li], who used it to derive an optimalHolder exponent for optimal maps under the strict condition (A3) on the cost, sharpening theHolder continuity result of Loeper [L1]. Furthermore, if (convex DASM) holds then −c(q, x)is convex in q for any x ∈ M , which then implies convexity of ϕ in q (see [FKM1, Theorem 4.3]for more details). One can easily check that c-segments with respect to x are transformed viathis coordinate change to ˜c-segments with respect to x, and c-segments with respect to x(q) aretransformed to c-segments with respect to q. Therefore, Loeper’s maximum principle (DASM)or (convex DASM) for c implies the same for c.

3.2.1 Relation between cotangent vectors in two different coordinates

Here we give an explicit relation between covectors in the new coordinate variable q (as intro-duced above) and the original coordinate variable x. Fix arbitrary y0 ∈ M , x0 ∈ M(y0), and letq0 = −Dxc(x0, y0) ∈ T ∗

y0M . For each z ∈ M(x0), consider the maps

z 7→ η(z) := −Dxc(x0, z) ∈ T ∗x0

M (3.1)z 7→ p(z) := −Dq c(q0, z) ∈ T ∗

q0(T ∗

y0M).

where c(q, x) := c(x(q), x)−c(x(q), y0) and the variables x and q are related as q(x) = −Dxc(x, y0).Denote by M∗(x0), M∗(q0) the embedding of M(x0) under the mappings z 7→ η(z), z 7→ p(z),respectively. These sets are related by an affine map as we see in the following lemma. Inparticular, from Assumption 2.3 both sets are convex in T ∗

x0M , T ∗

q0(T ∗

y0M), respectively.

Lemma 3.4. Let Assumptions 2.1 and 2.2 hold. Let η(p) denote the map from M∗(q0) toM∗(x0) that associates p(z) to η(z) as in the relation (3.1), and let η0 = −Dxc(x0, y0) ∈M∗(x0) ⊂ T ∗

x0M . Fix local coordinates. Then for all p ∈ M∗(q0), η(p) = (η(p)1, · · · , η(p)n)

is given asη(p)i = pj

(−DxiDxjc(x0, y0)

)+ ηi

0.

This formula allows the affine function p 7→ η(p) to be extended to a global map η : T ∗q0

(T ∗y0

M) →T ∗

x0M .

11

Proof. Observe that

ηi = −Dxic(x0, z)= Dxi

∣∣x=x0

[−c(x, z) + c(x, y0) − c(x, y0)]

= −Dqj

∣∣q=q0

[c(x(q), z) − c(x(q), y0)](Dxi

∣∣x=x0

qj) + ηi0

= pj (Dxi

∣∣x=x0

qj) + ηi0.

From the relationqj = −Dxjc(x(q), y0)

we see thatDxi

∣∣x=x0

q = −DxiDxjc(x0, y0),

and the assertion follows.

3.2.2 An estimate on the first derivatives of c

In Section 6.5 we will use the following simple estimate.

Lemma 3.5. Given convex sets Ω, Λ ⊂ Rn, assume that the function (q, y) ∈ Ω×Λ 7→ c(q, y) ∈ Ris smooth. Then for all q, q ∈ Ω and y ∈ Λ we have

| − Dqc(q, y) + Dqc(q, y)| ≤ C|q − q| |Dqc(q, y)|, (3.2)

where the constant C depends only on ‖c‖C3(Ω×Λ) and ‖(D2qyc)

−1‖L∞(Ω×Λ).

Proof. See [FKM1, Lemma 6.3].

4 An Alexandrov estimate: lower bound

In this section we show a key Alexandrov type estimate (4.1) which bounds from below the“oscillation” of φ inside a c-section by the measure of the section. (An estimate that comparethe oscillation of the function with the measure of the section is called Alexandrov type.) Thisresult is of its own interest, especially because it is proven under rather general assumptions,and does not rely on the special structure of products of spheres. In later sections, a companioninequality showing the upper bound will be obtained for a special choice of a c-section in theparticular case of products of spheres, see Theorem 6.4.

Lemma 4.1 (Alexandrov lower bound). Let M , M be complete n-dimensional Riemannianmanifolds. Suppose the cost c : M×M → R satisfies Assumptions 2.1, 2.2, 2.3 and 2.4 (convexDASM). Let φ be a c-convex function on M and assume 0 < λ ≤ |∂cφ| for a fixed λ ∈ R. Fix(x0, x0) ∈ M × M such that x0 ∈ ∂cφ(x0), and for h > 0 consider the c-section Zh defined as

Zh := x ∈ M | φ(x) − φ(x0) ≤ −c(x, x0) + c(x0, x0) + h.

Assume that −c(·, x0) is smooth on Zh, so that the function c-Exp−1x0

is defined and smooth onZh, or equivalently Zh ⊂ M(x0). Then the following inequality holds:

λ|Zh|2 ≤ C(n)[maxx∈Zh

|det(−DxDxc(x, x0))|minx∈Zh

|det(−DxDxc(x, x0))|

]2[supx∈Zh

supp′∈M∗(x)

|dc-Expx

∣∣p=p′

|]hn (4.1)

with the constant C(n) = (4n)n|B1|2, where |B1| denotes the measure of the unit ball in Rn.

12

Remark 4.2. In the statement of Lemma 4.1 and its proof, it is important to notice that by theassumption −c(·, x0) is smooth on clZh and Assumptions 2.1 and 2.2, the derivatives of c-Expx0

and its inverse (on Zh), i.e., −DxDxc(x, x0), x ∈ Zh, are all nonsingular.

Remark 4.3. For c = dist2 /2, Loeper’s maximum principle (DASM) (and so also (convexDASM)) implies that M = M has nonnegative sectional curvature (see [L1]). Therefore in thiscase c-Expy is a contraction, that is

supp′∈M∗(x)

|dc-Expx

∣∣p=p′

| ≤ 1.

We do not know if this contraction property holds for general non-negatively cross-curved costfunctions.

Remark 4.4. As in [FKM1], for our main results we will later follow the strategy developedin [Ca1] by using renormalization techniques, but only after a suitable change of coordinates.The main feature of our Alexandrov estimate with respect to the ones in [Ca1, FKM1] is inthe fact that the only “possibly bad” dependence on the cost function comes from the termmaxx∈Zh

|det(−DxDxc(x,x0))|minx∈Zh

| det(−DxDxc(x,x0))| , which can be made as close to 1 as desired, provided one can ensurethat the section Zh converges to a point as h → 0. This will play a crucial role in the proof ofTheorem 5.1, as it will allow us to apply this estimate near points which are arbitrarily close tothe cut-locus.

Proof. For globally smooth cost functions (on the products of two bounded domains) a similarresult was proved in [FKM1, Theorem 6.4]. In the present case where the cost function hassingularities, the previous proof does not work any more and we require the following subtleargument.

Consider the coordinate change x ∈ Zh 7→ q = −Dxc(x(q), x0) ∈ Wh ⊂ M∗(x0) ⊂ T ∗x0

M ,i.e., x = c-Expx0

q and Zh = c-Expx0(Wh), and let

mx(·) := −c(·, x) + c(·, x0).

As explained in Section 3.2, in these new coordinates the functions

q 7→ mx(x(q)) and q 7→ ϕ(q) = φ(x(q)) + c(x(q), x0)

are convex. Moreover the set Wh is convex, as

Wh = q ∈ T ∗x0

M | ϕ(q) − ϕ(q0) ≤ h,

where q0 is the point corresponding to x0 in the new coordinates, i.e., c-Expx0q0 = x0. It is also

important to notice that x0 ∈ ∂cφ(x0) implies ϕ(q)−ϕ(q0) ≥ 0. We now use Lemma 3.1 to findan affine map A : T ∗

x0M ' Rn 7→ T ∗

x0M ' Rn such that A(Wh) = Wh, with B1 ⊂ Wh ⊂ Bn.

Denote qb = A(0) and xb = c-Expx0qb. Define the renormalized function ϕ(q) := ϕ(Aq) for

each q ∈ Wh, and denote 12Wh := A(1

2Wh) and 12Zh := c-Expx0

(12Wh), where 1

2Wh denotes thedilation of Wh by a factor 1/2 with respect to the origin. This 1/2-dilation (or any factor in(0, 1) works) will be important in this proof.

Consider the reciprocal expression

∂cφ(1

2Zh

)= c-Expxb

(−Dxc(xb, x0) + V)

13

where

V :=

Dxmx(xb) | x ∈ ∂cφ(1

2Zh

)⊂ T ∗

xbM.

Here Dxmx denotes the differential when mx is differentiable, otherwise it means an arbitrarycovector in the subdifferential ∂mx(x). Notice that −Dxc(xb, x0) + V ⊂ cl(M∗(xb)), and thus∣∣∣∂cφ

(12Zh

)∣∣∣ ≤ (sup

p′∈M∗(xb)|dc-Expxb

∣∣p=p′

|)|V|. (4.2)

Now, the left-hand side is bounded from below as∣∣∣∂cφ(1

2Zh

)∣∣∣ ≥ λ∣∣∣12Zh

∣∣∣ (by the assumption |∂cφ| ≥ λ)

≥ λ[

minw∈Wh

|det(dc-Expx0

∣∣q=w

)|](1

2

)n|Wh| (by 1

2Zh = c-Expx0(12Wh))

≥ λminw∈Wh

|det(dc-Expx0

∣∣q=w

)|maxw∈Wh

|det(dc-Expx0

∣∣q=w

)|

(12

)n|Zh|

≥ λminx∈Zh

|det(−DxDxc(x, x0))|maxx∈Zh

|det(−DxDxc(x, x0))|

(12

)n|Zh| (by Dxc(·, x0) = c-Exp−1

x0).

(4.3)

In the following we will bound |V| from above by

maxx∈Zh|det(−DxDxc(x, x0))|

minx∈Zh|det(−DxDxc(x, x0))|

hn

|Zh|,

which will finish the proof; here the dilation 12Zh plays a crucial role (see (4.5)). Fix x ∈

∂cφ(12Zh), and let qx ∈ 1

2Wh such that x ∈ ∂ cϕ(qx). Here, the cost function c is the modifiedcost function accordingly with the coordinate changes:

c(q, y) := c(c-Expx0(Aq), y) − c(c-Expx0

(Aq), x0).

Consider the function

mx(q) := mx(c-Expx0(Aq)) = −c(q, x). (4.4)

Then

mx(q) − mx(qx) + ϕ(qx) ≤ ϕ(q) for q ∈ Wh.

We observe that mx(·) − mx(qx) is a convex function on Wh which vanishes at qx ∈ 12Wh, and

mx(·) − mx(qx) ≤ h on ∂Wh. Since B1 ⊂ Wh ⊂ Bn this easily gives mx(0) − mx(qx) ≥ −h,which by convexity implies

|Dqmx(0)| ≤ 2h. (4.5)

To get information on Dxmx, observe that from (4.4)

(dc-Expx0

∣∣q=qb

A)∗Dxmx(xb) = Dqmx(0), (4.6)

14

where (dc-Expx0

∣∣q=qb

A)∗ : T ∗xb

M 7→ T ∗0 (T ∗

x0M) is the dual map of the derivative map dc-Expx0

A :T0(T ∗

x0M) 7→ Txb

M . Here we abuse the notation and A denotes both the affine map and itsderivative. Moreover we use the canonical identification T ∗

0 (T ∗x0

M) ≈ T0(T ∗x0

M) ≈ T ∗x0

M . Hence(4.5) and (4.6) imply the key inclusion

V ⊂(dc-Expx0

∣∣∗q=qb

)−1(A∗)−1B2h,

so that

|V| ≤∣∣ det(dc-Expx0

∣∣∗q=qb

)−1∣∣∣∣ det(A∗)−1

∣∣|B1|2nhn

= |det(dc-Expx0

∣∣q=q0

)|−1|det A|−1|B1|2nhn

(by the identification between vectors and covectors)

≤ |det(dc-Expx0

∣∣q=q0

)|−1|B1|2(2n)n hn

|Wh|(by |Wh| = |det A||Wh| ≤ | detA||B1|nn)

≤maxw∈Wh

|det(dc-Expx0

∣∣q=w

)||det(dc-Expx0

∣∣q=q0

)||B1|2(2n)n hn

|Zh|

(by |Zh| ≤ maxw∈Wh|det(dc-Expx0

∣∣q=w

)||Wh| )

≤ maxx∈Zh|det(−DxDxc(x, x0)|)

minx∈Zh|det(−DxDxc(x, x0))|

|B1|2(2n)n hn

|Zh|(by Dxc(·, x0) = c-Exp−1

x0).

Together with (4.2) and (4.3), this concludes the proof.

5 Main result: Stay-away property on multiple products ofspheres

From now on we restrict our attention to the case M = M = M1 × . . . × Mk, where for eachi = 1, . . . , k, M i = Sni

riis a round sphere of constant sectional curvature r−2

i . Though M = M ,we sometimes keep the bar notation to emphasize the distinction between the source and thetarget domain of the transportation. Let x = (x1, . . . , xk) and x = (x1, . . . , xk) denote points inthe product M1 × . . .×Mk, with xi, xi ∈ M i, i = 1, . . . , k. Assume that the transportation costc on M is the tensor product of the costs ci on each M i, defined as

c(x, x) :=k∑

i=1

ci(xi, xi). (5.1)

Assume moreover that each ci is of the form f i(disti) (disti being the distance on M i) for somesmooth strongly convex even function f i : R → R, normalized so that f i(0) = 0. (This normal-ization assumption can be done with no loss of generality, as one can always add an arbitraryconstant to the cost function.) Moreover we suppose that each ci satisfies Assumptions 2.1, 2.2,2.3, 2.4 and 2.5 in Section 2. As shown in [KM2], under these assumptions the tensor productcost c also satisfies Assumptions 2.1, 2.2, 2.3 and 2.4 (but not necessarily 2.5). The reader shouldhave in mind that our model example is c = dist2 /2, which as shown in [KM2] satisfies all theassumptions above. However we prefer to give a proof of the result with general f i since this willnot cost further effort in the proof, and we believe it may be of interest for future applications.

15

Let us observe that for any point x we have M(x) = M1(x1) × . . . × Mk(xk) and M∗(x) =M∗(x1) × . . . × M∗(xk). Moreover, since the distance squared function on a round sphere issmooth except for antipodal pairs, for each xi ∈ M i we have Cut(xi) = −xi, where −xi

denotes the antipodal point of xi. (We also write −x = (−x1, · · · ,−xk).) This implies easilythat c-Cut(x) = Cut(x), so that M i(xi) = M i \ −xi and c-Cut(x) is a union of (totallygeodesic) submanifolds, each of which is an embedding of a product M i1 × . . . × M il , l < k.

The goal of the rest of the paper is to show a stay-away property of optimal transport mapson products of spheres:

Theorem 5.1 (Stay-away from cut-locus). Let M = M = M1 × . . . × Mk, where for eachi = 1, . . . , k, M i = Sni

riis a round sphere of constant sectional curvature r−2

i . Let c be the costgiven in (5.1) with ci is of the form f i(disti), where f i : R → R are smooth strongly convex evenfunctions such that f i(0) = 0. Assume further that each cost ci satisfies Assumptions 2.1, 2.2,2.3, 2.4 and 2.5, and let φ be a c-convex function satisfying Assumption 2.10. Then

∂cφ(x) ∩ c-Cut(x) = ∅ ∀x ∈ M.

Equivalently, for every x ∈ M the contact set S(x) = ∂ cφ(x) satisfies

S(x) ∩ c-Cut(x) = ∅.

Before sketching the proof of this result, let us first see its consequences:

Corollary 5.2 (Uniformly stay-away from cut-locus). Use the notation and assumptionsas in Theorem 5.1. There exists a positive constant C depending only on λ (see Assumption 2.10)and ni, ri, f i, for i = 1, · · · , k, such that

dist(∂cφ(x), c-Cut(x)

)≥ C ∀x ∈ M.

where dist denotes the Riemannian distance of M .

Proof. The result follows by compactness. Indeed, suppose by contradiction there exists asequence of c-convex functions φl satisfying Assumption 2.10, and xl ∈ M such that

dist(∂cφl(xl), c-Cut(xl)

)→ 0 as l → ∞.

Up to adding a constant, we can also assume that φl(xl) = 0. Then, since M is compact and thefunctions φl are uniformly semiconvex (and so uniformly Lipschitz), applying Arzela-Ascoli’sTheorem, up to a subsequence there exists a c-convex function φ∞ and x∞ ∈ M such thatφl → φ∞ uniformly and xl → x∞. We now observe that also φ∞ satisfies Assumption 2.10(see for instance [FKM1, Lemma 3.1]). Moreover, by the definition of c-subdifferential we easilyobtain

yl ∈ ∂cφl(xl), yl → y∞ ⇒ y∞ ∈ ∂cφ∞(x∞).

This impliesdist

(∂cφ∞(x∞), c-Cut(x∞)

)= 0,

which contradicts Theorem 5.1, and completes the proof.

Corollary 5.3 (Regularity of optimal maps). Let M, M, c be as in Theorem 5.1. Assumethat µ and ν are two probability measures absolutely continuous with respect to the volumemeasure, and whose densities are bounded away from zero and infinity. Then the unique optimalmap T from µ to ν is injective and continuous. Furthermore, if both densities are Cα/C∞, thenT is C1,α/C∞.

16

Remark 5.4. The C1,α-regularity result (C2,α for the potential φ) in this corollary is a directconsequence of the injectivity and continuity of T applied to the theory of Liu,Trudinger andWang [LTW]. The higher regularity C∞ follows from Schauder estimates.

Proof. We recall that, under the assumption that µ and ν have densities bounded away fromzero and infinity, there exists a c-convex function φ such that T (x) = c-Expx(∇φ(x)) a.e., andφ satisfies Assumption 2.10 (see for instance [MTW] or [FKM1, Lemma 3.1]). Hence it sufficesto prove that φ is C1 and strictly c-convex, in the sense that S(x) = ∂ cφ(x) is a singleton forevery x ∈ ∂cφ(M) = M .

To this aim, we observe that once we know that φ is strictly c-convex, then we can localizethe proof of the C1 regularity in [FKM1] to obtain the desired result. Thus we only need toshow the strict c-convexity of φ.

Fix x ∈ M . By Theorem 5.1 we know that S(x) ⊂ M(x), so that in a neighborhood ofS(x) we can consider the change of coordinates x 7→ q = −Dc(x, x) ∈ T ∗

xM . As shown in[FKM1, Theorem 4.3], thanks to Loeper’s maximum principle (DASM) the set S(x) is convexin these coordinates. Moreover, since now the cost is smooth in a neighborhood of S(x), by[FKM1, Theorem 7.1 and Remark 7.2] the compact convex set S(x) in the new coordinates hasno exposed points on the support of |∂cφ|.2 Since in our case the support of |∂cφ| is the wholeM , the only possibility left is that S(x) is a singleton, as desired.

Sketch of the proof of Theorem 5.1. We prove this theorem by contradiction. Assume there ex-ists a point x0 such that the contact set S(x0) intersects Cut(x0). First, we find a cut-exposedpoint x0 in S(x0) ∩ Cut(x0). More precisely we split M as M · × M ·· so that x0 = (x·

0, x··0),

x0 = (x·0, x

··0), where x·

0 = −x·0 ∈ Cut(x·

0), x··0 stays away from the cut-locus of x··

0 , and x··0 is an

exposed point in the set S·· = y·· ∈ M ·· | (−x·0, y

··) ∈ S(x0) (see Section 6.1). Near x0, forε ∈ (0, 1) and δ ∈ [0, 1] we construct a family of points xε,δ = (x·

ε, x··δ ) such that d(x0, xε,δ) ≈ ε+δ,

so that for δ small we have xε,δ ∈ M∗(x0), or equivalently x0 ∈ M∗(xε,δ). By suitably choosingthe point x··

δ in order to exploit the fact that x··0 is an exposed point for S··, we can ensure that,

if Zε,δ,h denotes a section obtained by cutting the graph φ with −c(·, xε,δ) at height h above x0,then for any fixed ε ∈ (0, 1) we have Zε,δ,h → x0 as δ, h

δ → 0 (see Section 6.2). In particular,for ε > 0 fixed we have Zε,δ,h ⊂ M∗(xε,δ) for δ, h

δ small (equivalently, the function c(·, xε,δ) issmooth inside Zε,δ,h). Now we take advantage of the choice of x·

ε: on the sphere Sn the func-tion −dist2(·, x) looks like a cone near the antipodal point −x, and if dist(x, xε) ≈ ε then themeasure of a section obtained by cutting the graph of −dist2(·, x) with − dist2(·, xε) at height habove −x has measure ≈ hn/ε (see Proposition 6.7). In our case, since x·

0 = −x·0, the function

φ behaves as −c(·, x0) ≈ −dist(·, x0) along M · (see Lemma 6.6). Hence by the argument abovewe have an improvement of a factor 1/ε in the measure of Zε,δ,h (see Proposition 6.7), whichallows to show the following Alexandrov type inequality:

hdim M . ε|Zε,δ,h||∂cφ(Zε,δ,h)| for δ and hδ sufficiently small,

where . is independent of ε, δ and h (see Theorem 6.4)3. Thanks to Assumption 2.10, the above2Recall that, given a compact convex set Z ⊂ Rn, z ∈ ∂Z is an exposed point if there exists an hyperplane

Π ⊂ Rn such that Z ∩ Π = z.3Although this is the informal idea, the actual proof is much more involved. In particular, for technical reasons,

we will also need to split M as M ′ ×M ′′ so that ∂cφ(x0)∩`

−x′0×M ′′´ 6= ∅ (M ′ corresponds to the “cut-locus

components”) and ∂cφ(x0) ∩`

M ′ × −x′′0

´

= ∅ (M ′′ corresponds to the “regular components”), see Section 6.1.Observe that M · ⊂ M ′. Then, to prove Theorem 6.4, a different argument has to be used depending on the kindof components, see Sections 6.4 and 6.5 respectively.

17

_ xε_

_ x_

hn/ε~~

h

ε~~

_ dist (., xε)_

2

_ dist (., x )_2

Figure 1: On the sphere, the squared distance function from a point x looks like a cone near −x. So, ifdist(x, xε) = dist(−x,−xε) ≈ ε, the section obtained by cutting its graph with −dist2(·, xε) at height h

has measure ≈ hn/ε.

inequality implies

hdim M . ε

λ|Zε,δ,h|2 for δ and h

δ sufficiently small. (5.2)

On the other hand, since Zε,δ,h ⊂ M∗(xε,δ) for δ and hδ small enough, we can apply Lemma 4.1

to Zε,δ,h and have

λ|Zε,δ,h|2 ≤ C(n)[

maxx∈Zε,δ,h|det(−DxDxc(x, xε,δ)|

minx∈Zε,δ,h|det(−DxDxc(x, xε,δ))|

]2[sup

x∈Zε,δ,h

supp′∈M∗(x)

|dc-Expx

∣∣p=p′

|]hdim M .

The convergence Zε,δ,h → x0 as δ, hδ → 0 further reduces this inequality to

λ|Zε,δ,h|2 . hdim M for δ and hδ sufficiently small,

which contradicts (5.2) as ε → 0 and completes the proof.

The rest of the paper is devoted to fleshing out the details of the above proof.

6 Proof of Theorem 5.1 (Stay-away from cut-locus)

6.1 Cut-exposed points of contact sets

Assume by contradiction that there exists x0 = (x10, . . . , x

k0) ∈ M = M = M1 × . . . × Mk such

that S(x0) ∩ c-Cut(x0) 6= ∅. To prove Theorem 5.1 a first step is to find a cut-exposed point ofthe contact set in the intersection with the cut-locus, which we define throughout the presentsection.

Let y ∈ S(x0) ∩ c-Cut(x0), and note that one of the components of y = (y1, . . . , yk), say yj ,satisfies yj = −xj

0. Moreover we cannot have y = −x0. Indeed it is not difficult to see that, ifx0 ∈ ∂cφ(−x0), then ∂cφ(−x0) = M (see for instance Lemma 6.6(1) below), which contradictsAssumption 2.10.

Among all points y ∈ S(x0)∩ c-Cut(x0), choose one such that the number a0 of its antipodal(or cut-locus) components is maximal, and denote the point by y0. By rearranging the productM1 × . . . × Mk, we may write without loss of generality that

y0 = (−x10, . . . ,−xa0

0 , ya0+10 , . . . , yk

0 ), yj0 6∈ c-Cut(xj

0) ∀ j = a0 + 1, . . . , k. (6.1)

18

For convenience, use the expression

M · = M · = M1 × . . . × Ma0 , M ·· = M ·· = Ma0+1 × . . . × Mk.

The expressions A·, A·· will be used to denote things defined for elements in M ·, M ··, respectively.For example,

y· = (y1, . . . , ya0), y·· = (ya0+1, . . . , yk),

c·(y·, y·) =a0∑i=1

ci(yi, yi), c··(y··, y··) =k∑

i=a0+1

ci(yi, yi).

Consider the set

S·· = y·· ∈ M ·· | (−x·0, y

··) ∈ S(x0).

Notice that due to maximality of a0, S·· ⊂ M ··(x··0) and it is embedded to M ··∗(x··

0) through themap y·· 7→ q··(y··) = −Dx··c··(y··, x··

0). Observe that since S·· is compact, the resulting set, sayS··, is compact too. Moreover S·· is convex since it is the restriction of the convex set S(x0) toM ··∗(x··

0), where S(x0) is the image of S(x0) under the map x 7→ −Dxc(x, x0). (More precisely,this set S(x0) is defined as the closure of the image of S(x0)\c-Cut(x0).) This compact convexityensures the existence of an exposed point q··0 for S··, that is, there exists an affine function L onT ∗

x··0M ·· such that

L(q··0 ) = 0, and L(q··) < 0 ∀q·· ∈ S·· \ q··0. (6.2)

(In case S·· = q··0 let L ≡ 0.) One should note that if L is such an affine function, then tL isalso such an affine function for any t > 0. Let x0 ∈ S(x0) be the corresponding point of q··0 inM , that is,

x0 = (−x·0, x

··0), (6.3)

where

q··0 = −Dx··c··(x··0 , x

··0).

We call this point x0 a cut-exposed point of S(x0), since its components are either cut-locus typeor exposed.

One can assume with a further rearrangement of the product M ·· = Ma0+1 × . . .×Mk thatthere exists b0 ∈ a0, . . . , k with the following two properties:

1. For each i1 ∈ a0 + 1, . . . , b0, there exists yi1 ∈ ∂cφ(x0) with

yi1i1

= −xi10 . (6.4)

2. For every y ∈ ∂cφ(x0),

yj 6= −xj0 (or equivalently yj /∈ c-Cut(xj

0) = Cut(xj0)) for j = b0 + 1, . . . , k.

(6.5)

(If b0 = a0, a0 + 1, . . . , b0 = ∅.)

19

M..

S..

M.

S(x0)_

x0

__ .

y0

x0

Figure 2: Starting from a point y0 ∈ S(x0) such that the number of its antipodal (or cut-locus) compo-nents is maximal, we choose x0 = (y·

0, x··0 ) = (−x·

0, x··0 ) ∈ M · × M ·· = M so that x··

0 is an exposed pointfor S·· (in some suitable system of coordinates).

After this rearrangement, define

M ′ = M ′ = M1 × . . . × M b0 M ′′ = M ′′ = M b0+1 × . . . × Mk

The expressions A′, A′′ will be used to denote things defined for elements in M ′, M ′′, respectively.For example,

y = (y′, y′′), y = (y′, y′′) ∈ M = M ′ × M ′′ = M ′ × M ′′,

c(y, y) = c′(y′, y′) + c′′(y′′, y′′),

and we have the identification

T ∗x0

M = T ∗x′0M ′ × T ∗

x′′0M ′′, M∗(x0) = M ′∗(x′

0) × M ′′∗(x′′0).

In the following n′ = dim M ′, n′′ = dim M ′′ and π′, π′′ denote the canonical projections fromM to M ′, M ′′, respectively. This splitting of M as M ′ × M ′′ will be important later, as in theproof of Theorem 6.4 we will need different arguments on M ′ and M ′′ respectively, see Sections6.4 and 6.5.

6.2 Analysis near the cut-exposed point

In this subsection we construct a family of c-sections Zε,δ,h of φ near the cut-exposed point x0

defined in (6.3). Regarding these c-sections, two important results (Proposition 6.2 and 6.3)are obtained. In later subsections we will show an Alexandrov type inequality for Zε,δ,h whichwill be paired with the other Alexandrov type inequality (4.1) to lead a contradiction to theexistence of such x0, thus finishing the proof of Theorem 5.1.

Recall the affine function L on T ∗x··0M ·· given in (6.2). After modifying L by multiplying it

by an appropriate positive constant, there exists a geodesic curve [0, 1] 3 δ 7→ x··δ ∈ M ··(x··

0)starting from x··

0 such that for the linear map ∇L on T ∗x··M ··,

∇L(q·· − q··0 ) = 〈 ∂

∂t

∣∣∣t=0

x··t , q·· − q··0 〉. (6.6)

20

Consider a c·-segment [0, 1] 3 ε → x·ε ∈ M · with respect to x·

0 connecting the point x·0 to its

antipodal point x·1/2 = x·

0 then to x·1 = x·

0. (xε is nothing else than a closed geodesic startingfrom x·

0 and passing through x·0 = −x·

0 at ε = 1/2.) Define

xε,δ := (x·ε, x

··δ ) ∈ M = M · × M ··. (6.7)

Obviously x0,0 = x0. Two important properties follow:

(a) Since x·ε ∈ M ·(x·

0) for ε ∈ (0, 1) and x··0 ∈ M ··(x··

0) we have

x0 ∈ M(xε,δ) = M ·(x·ε) × M ··(x··

δ ) ∀ 0 < ε < 1, δ ≥ 0 small.

(b) Since x·ε,δ = x·

ε 6= x·0 for ε ∈ (0, 1), x·

1 = x·0, and x·

0 = −x·0, for every ε ∈ (0, 1) and

δ ∈ [0, 1] we have from Assumption 2.5 for each ci,

−c·(x·, x·ε,δ) + c·(x·

0, x·ε,δ) ≤ −c·(x·, x·

0) + c·(x·0, x

·0) ∀x· ∈ M ·, (6.8)

with equality only when x· = x·0. (See for instance Lemma 6.5 below.)

Consider now the c-section Zε,δ,h obtained by cutting the graph of φ by the graph of−c(·, xε,δ) + c(x0, xε,δ) + h, that is

Zε,δ,h := x ∈ M | φ(x) − φ(x0) + c(x, xε,δ) − c(x0, xε,δ) ≤ h. (6.9)

As it can be easily seen by moving down the graph of −c(·, xε,δ) and lift it up until it touchesthe graph of φ, xε,δ ∈ ∂φ(Zε,δ,h). Hence, thanks to Loeper’s maximum principle (DASM) wehave

xε,δ ∈ ∂cφ(Zε,δ,h). (6.10)

Proposition 6.1. The following equality holds.

Zε,0,0 = S(xε,0) = S(x0) ∩(x·

0 × M ··). (6.11)

Proof. From (6.8),

φ(x) − φ(x0) + c(x, xε,0) − c(x0, xε,0)= φ(x) − φ(x0) + c·(x·, x·

ε,0) − c·(x·0, x

·ε,0) + c··(x··, x··

0) − c··(x··0 , x

··0)

≥ φ(x) − φ(x0) + c·(x·, x·0) − c·(x·

0, x·0) + c··(x··, x··

0) − c··(x··0 , x

··0)

= φ(x) − φ(x0) + c(x, x0) − c(x0, x0) ≥ 0.

This, together with the equality case for (6.8), yields (6.11).

The following two propositions are essential in our proof of Theorem 5.1.Our first result shows that, for fixed ε, the sections Zε,δ,h converge to the point x0 (in the

sense of Kuratowski) as δ, hδ → 0.

Proposition 6.2. Fix 0 < ε < 1. Then, for any sequences δi, hi → 0 with hiδi

→ 0, we have

Zε,δi,hi→ x0 as i → ∞.

21

Proof. Fix arbitrary sequences δi,hiδi

→ 0, and denote

Z∞ = limi→∞

Zε,δi,hi= z∞ ∈ M | there exists a sequence zi ∈ Zε,δi,hi

with zi → z∞ ∈ M.

By continuity, z∞ ∈ Zε,0,0 for each z∞ ∈ Z∞, and thus by (6.11) z·∞ = x·0.

To show z··∞ = x··0 , we first let δ > 0 be sufficiently small and fix a small (closed) neighborhood,

say U , of x0 so that all the derivatives (up to the second order) of the function U×[0, 1] 3 (x, t) →c(x, xε,tδ) are uniformly bounded. Then, for x ∈ Zε,δ,h ∩ U the following inequalities hold:

h ≥ φ(x) − φ(x0) + c·(x·, x·ε) − c·(x·

0, x·ε) + c··(x··, x··

δ ) − c··(x··0 , x

··δ )

≥ −c·(x·, x·0) + c·(x·

0, x·0) − c··(x··, x··

0) + c··(x··0 , x

··0)

+ c·(x·, x·ε) − c·(x·

0, x·ε) + c··(x··, x··

δ ) − c··(x··0 , x

··δ ) (since x0 ∈ ∂cφ(x0))

≥ −c··(x··, x··0) + c··(x··

0 , x··0) + c··(x··, x··

δ ) − c··(x··0 , x

··δ ) (by (6.8))

≥ 〈Dxc··(x··, x··0) − Dxc··(x··

0 , x··0),

∂

∂t

∣∣∣t=0

xtδ〉 + O(δ2)

= δ∇L(Dxc··(x··, x··0) − Dxc··(x··

0 , x··0)) + O(δ2) (by (6.6) )

Use the coordinate q··(x··) = −Dx··c··(x··, x··0) to rewrite this as

∇L(q·· − q··0 ) ≥ −h

δ− O(δ).

Since L(q··0 ) = 0 this gives

L(q·· − q··0 ) ≥ −h

δ− O(δ).

Consider now the sequences δi,hiδi

→ 0, and any convergent subsequence of zi ∈ Zε,δi,hi∩U . For

the limit z∞, let q·· = −Dx··c(z··∞, x··0). Then q··∞ ∈ S·· (since z∞ ∈ Zε,0,0 ⊂ S(x0) by (6.11)), and

from the above inequality we get

L(q··∞ − q··0 ) ≥ 0

which forces q··∞ = q··0 by (6.2). This shows z··∞ = x··0 , and thus Z∞ ∩ U = x0. To finish the

proof notice that each Zε,δ,h is connected, and so is the limit Z∞. (This connectivity can beseen by noticing that the set Zε,δ,h is convex in the coordinates q(x) = −Dxc(x, xε,δ) ∈ T ∗

xε,δM .)

Therefore Z∞ = x0, as desired.

Proposition 6.3. There exists δ0 = δ0(ε) > 0 such that, if 0 ≤ δ ≤ δ0, 0 ≤ h ≤ δ2, then for eachy = (y′, y′′) ∈ ∂cφ(Zε,δ,h) the component y′′ stays away from the cut-locus of the component z′′

of z (i.e., y′′ ∈ M ′′(z′′)) for every z ∈ Zε,δ,h. Equivalently π′′(∂cφ(Zε,δ,h)) ⊂∩

z∈Zε,δ,hM ′′(z′′).

Proof. Suppose the statement is false along some sequence δi, hi → 0 with hi ≤ δ2i , and let

xi, zi ∈ Zε,δ,h, yi ∈ ∂cφ(xi) be such that y′′i ∈ Cut(z′′i ). Since Zε,δi,hi→ x0, both xi, zi → x0.

Moreover if y∞ is a cluster point for yii∈N, then y∞ ∈ ∂cφ(x0) and y′′∞ ∈ Cut(x′′0). This

contradicts the choice of M ′′ (see (6.5)) and concludes the proof.

22

6.3 An Alexandrov type estimate near the cut-exposed point

We state the main theorem to be proved in the rest of the paper.

Theorem 6.4 (Alexandrov upper bound near cut-exposed point). Fix 0 < ε < 1, andlet Zε,δ,h be as in (6.9). There exists δ1 = δ1(ε) > 0 so that, if 0 < δ ≤ δ1, then there existsh1 = h1(ε, δ) such that

hdim M . εa0 |Zε,δ,h||∂cφ(Zε,δ,h)| ∀ 0 < h ≤ h1(ε, δ), (6.12)

where . is independent of ε, δ and h.

This result concludes the proof of Theorem 5.1, since for ε > 0 small enough and δ, hδ → 0

we have Zε,δ,h → x0 (by Proposition 6.2), and (6.12) is in contradiction with (4.1).The following subsections are devoted to the proof of Theorem 6.4, that we divide into

three parts. First, in Section 6.4 we get Alexandrov type estimates for the sets obtained bythe intersection of Zε,δ,h with the cut-locus components of x0. In Section 6.5, we analyze theprojection π′′(Zε,δ,h) of Zε,δ,h onto the regular component M ′′ of x0. We construct a suitableconvex set, say C, which has size comparable to the image ∂cφ(Zε,δ,h), and we get a version ofthe estimate (6.12) involving C and π′′(Zε,δ,h) (see Proposition 6.8(3)). Finally in Section 6.6we combine these results and conclude the proof.

6.4 Proof of Theorem 6.4 (Alexandrov upper bound near cut-exposed point):analysis in the cut-locus component M ′

The main result of this section is Proposition 6.7 that gives an Alexandrov type estimate for theintersection of Zε,δ,h with the cut-locus components of x0.

We start with a few elementary results.

Lemma 6.5. Let Sn be the standard round sphere, and c(x, x) = f(dist(x, x)) for (x, x) ∈Sn × Sn, where f is a smooth strictly increasing function f : R+ → R+. Assume that c satisfiesAssumption 2.5 (DASM+). Then, for every x, x ∈ Sn,

−c(−x, y) + c(−x, x) ≥ −c(x, y) + c(x, x) ∀ y ∈ Sn,

where −x denotes the antipodal point of x. Moreover equality holds if and only if y = x.

Proof. For any x, x ∈ Sn, one can find a c-segment x(s) with respect to x such that x(0) = x(1) =−x and x(s0) = x for some s0 ∈ [0, 1]. The inequality (together with the characterization of theequality case) then follows from (DASM+) for the function ms(·) = −c(x(s), ·)+ c(x(s), x).

For each 1 ≤ i ≤ k and z ∈ M , let M iz denote the i-th slice of M through z, that is

M iz := x ∈ M | xj = zj for j 6= i.

The following lemma generalizes the fact that on M = M = Sn with c = dist2 /2, if x ∈ Sn and−x ∈ ∂cφ(x), then ∂cφ(x) = Sn.

Lemma 6.6. Let M, M, c be as in Theorem 5.1. Let φ be a c-convex function on M . Fixz = (z1, . . . , zk) ∈ M = M1 × . . . × Mk and an open set U with z ∈ U . Fix i ∈ 1, . . . , k, andlet z ∈ M with zi = −zi. The following holds:

23

(1) If z ∈ [∂cφ(U)]z (resp. z ∈ ∂cφ(z)), then M iz ⊂ [∂cφ(U)]z (resp. M i

z ⊂ ∂cφ(z)).

(2) Suppose z ∈ ∂cφ(z). Then, for each x ∈ M iz, φ(x) − φ(z) = −ci(xi,−zi) + ci(zi,−zi).

Proof. To prove (1) it is enough to observe that for x ∈ M iz and x ∈ M ,

− c(x, x) + c(z, x)

= −ci(xi, xi) + c(zi, xi) +∑j 6=i

[− cj(xj , zj) + cj(xj , zj)

]≤ −ci(xi,−zi) + c(zi,−zi) +

∑j 6=i

[− cj(xj , zj) + cj(zj , zj)

](by Lemma 6.5)

= −c(x, z) + c(z, z) (since zi = −zi).

The last line is bounded from above by φ(x) − φ(z) either if x ∈ ∂U or z ∈ ∂cφ(z).Let us prove the (2). Suppose z ∈ ∂cφ(z). By duality (Lemma 2.6), z ∈ ∂ cφ(z) for the

dual c-convex function φ. Applying (1) to φ we get M iz ∈ ∂ cφ(z), or equivalently M i

z ⊂ S(z).Therefore for all x ∈ M i

z we have

φ(x) − φ(z) = −c(x, z) + c(z, z)

= −ci(xi,−zi) + ci(zi,−zi) (since xj = zj for j 6= i)

which concludes the proof.

Let i ∈ 1, . . . , b0, i.e., M i is a component of M ′. Recall that x0 is the cut-exposed pointdefined in (6.3). By definition of b0 in (6.4) and (6.5), there exists yi ∈ ∂cφ(x0) such thatyi

i = −xi0. (If i ≤ a0 then one can choose yi = x0.) Let Zi

ε,δ,h := πi(Zε,δ,h ∩ M i

x0

)for the

canonical projection πi : M → M i. Then Lemma 6.6(2) implies

Ziε,δ,h = xi ∈ M i | − ci(xi,−xi

0) + ci(xi0,−xi

0) + ci(xi, xiε,δ) − ci(xi

0, xiε,δ) ≤ h. (6.13)

Here comes the main result of this section.

Proposition 6.7. There exist δ2 = δ2(ε) > 0 such that, if 0 < δ ≤ δ2, then there existsh2 = h2(ε, δ) such that the set Zi

ε,δ,h satisfies the following estimates for 0 < h ≤ h2:

hdim M i . ε|Ziε,δ,h| if 1 ≤ i ≤ a0,

hdim M i . |Ziε,δ,h| if a0 + 1 ≤ i ≤ b0,

where . is independent of ε, δ and h and |Ziε,δ,h| denotes the Riemannian volume in the sub-

manifold M i.

Proof. From (6.13) and Lemma 6.5 we have Ziε,δ,h → xi

0 as h → 0. Thus for sufficiently smallh we can embed Zi

ε,δ,h into ∈ T ∗xi0M i by xi 7→ qi(xi) = −Dxici(xi, xi

0). Let W ih be its image.

Then

|W ih| ≤

(max

xi∈Ziε,δ,h

∣∣−DxiDxici(xi, xi0)

∣∣) |Ziε,δ,h| . |Zi

ε,δ,h|

for h sufficiently small. In the following we bound |W ih| from below.

24

Without loss of generality, assume M i is the unit sphere. Let qi0 = qi(xi

0). By abuse ofnotation use ci(qi, xi) to denote ci(xi(qi), xi), and renormalize this cost function as

cih(qi, xi) =

1h

[ci(hqi + qi

0, xi) − ci(qi

0, xi)

]Then (6.13) implies W i

h = hW ih + qi

0, where

W ih := qi ∈ T ∗

xi0M i | − ci

h(qi,−xi0) + ci

h(qi, xiε,δ) ≤ 1

Recall ci = f i(disti) for some smooth nonnegative uniformly convex function f i : R+ → R+

such that f i(0) = 0, df i

dt (0) = 0. Thus, as h → 0 the renormalized cost −cih(qi,−xi

0) convergesto the conical function

qi 7→ df i

dt(π)|qi|, for qi ∈ T ∗

xi0M i.

(Here, we used disti(xi0,−xi

0) = π.)Case I: If 1 ≤ i ≤ a0, then xi

ε,δ = x· iε , and so ci

h(q, xiε,δ) converges to the linear function

qi 7→ Dqci(qi

0, x· iε ) · qi

where

|Dqci(qi

0, x· iε )| =

df i

dt(π − 2πε) ≥ df i

dt(π) − Cε

for some constant C > 0. (Here, we used dist(xi0, x

· iε ) = π − 2πε.) Therefore in the limit h → 0

one can easily check that

1 < i < a0_ _ a0 < i < b0

_

ε

1 1

> 1/ε∼ > 1∼

ch(., xi )+1iε,δ

_ _

ch(., xi )i_ _0 ch(., xi )i_ _

0

ch(., xi )+1iε,δ

_ _

iWh

iWh

~~

Figure 3: If 1 ≤ i ≤ a0 then −xi0 = xi

0 and disti(−xi0, x

iε,δ) ≈ ε, so the size of the section is of order 1/ε

(see also Figure 1). On the other hand, if a0 < i ≤ b0 then −xi0 6= xi

0, which implies that disti(−xi0, x

iε,δ)

is uniformly bounded away from 0, and the size of the section is of order 1.

limh→0

|W ih| & 1

ε,

and thus for h > 0 sufficiently small

|W ih| = hdim M i |W i

h| & hdim M i

ε.

Case II: If a0 < i ≤ b0, then xiε,δ = x·· i

δ . Similarly as for the above case, cih(q, xi

ε,δ) convergesto the linear function

qi 7→ Dqci(qi

0, x·· iδ ) · qi.

25

Since x··δ ∈ M ··(x··

0) for δ > 0 small enough, there exist positive constants C,C0 such that

|Dqci(qi

0, x·· iδ )| ≤ df i

dt(π − C) ≤ df i

dt(π) − C0,

where for the last inequality we used the uniform convexity of f . From this one can check thatlimh→0 |W i

h| & 1, and thus for sufficiently small h > 0

|W ih| & hdim M i

.

This concludes the proof.

6.5 Proof of Theorem 6.4 (Alexandrov upper bound near cut-exposed point):analysis in the regular component M ′′

The main result of this subsection is Proposition 6.8. Fix 0 < ε < 1, and assume that δand h

δ are sufficiently small so that, as in Proposition 6.3, the set Zε,δ,h is close to the cut-exposed point x0, and so in particular Zε,δ,h ⊂ M(xε,δ). Consider the change of coordinatesq ∈ T ∗

xε,δM 7→ x(q) ∈ M(xε,δ) induced by the relation

q = −Dxc(x(q), xε,δ), (6.14)

and let Zε,δ,h ∈ T ∗xε,δ

M be the set Zε,δ,h in this chart. The function φ and the cost c aretransformed to

ϕ(q) = φ(x(q)) + c(x(q), xε,δ),

andc(q, y) = c(x(q), y) − c(x(q), xε,δ) for (q, y) ∈ T ∗

xε,δM × M.

Notice thatc(q, xε,δ) ≡ 0,

and ϕ is a c-convex function on T ∗xε,δ

M . Moreover

Zε,δ,h = q ∈ T ∗xε,δ

M | ϕ(q) − ϕ(q0) ≤ h

where q0 is the point corresponding to x0 in this new chart. It is important to recall that, thanksto Assumption 2.4 (convex DASM), c and ϕ are convex. (See Section 3.2)

We have the natural decomposition (with obvious notation)

q = (q′, q′′) = (−Dx′c′(x′(q′), x′ε,δ),−Dx′′c′′(x′′(q′′), x′′

ε,δ)) (6.15)

∈ T ∗xε,δ

M = T ∗x′

ε,δM ′ × T ∗

x′′ε,δ

M ′′.

(Here, one should keep in mind that, by the definition of xε,δ, the component x′′ε,δ in M ′′ does

not depend on ε.) The modified cost c(q, y) has the decomposition

c(q, y) = c′(q′, y′) + c′′(q′′, y′′)

wherec′(q′, y′) = c′(x′(q′), y′) − c′(x′(q′), x′

ε,δ) for q′ ∈ T ∗x′

ε,δM ′,

26

andc′′(q′′, y′′) = c′(x′′(q′′), y′′) − c′′(x′′(q′′), x′′

ε,δ) for q′′ ∈ T ∗x′′

ε,δM ′′.

Let π′, π′′ denote the canonical projection from T ∗xε,δ

M onto T ∗x′

ε,δM ′ and T ∗

x′′ε,δ

M ′′, respectively.

Now, let us construct a convex set C ⊂ T ∗q0

(T ∗xε,δ

M) that we will use later to estimate|∂cφ(Zε,δ,h)| from below (see Proposition 6.9). The strategy of the proof follows the lines of theone of [FKM1, Proposition 6.10].

Proposition 6.8. Fix 0 < ε < 1, and assume that 0 < δ ≤ δ0 and 0 < h ≤ δ2, with δ0 asin Proposition 6.3. Then there exists a convex set C ∈ T ∗

q0(T ∗

xε,δM) satisfying the following

properties:

(1) C ⊂ 0 × T ∗q′′0

(T ∗x′′

ε,δM ′′) ⊂ T ∗

q′0(T ∗

x′ε,δ

M ′) × T ∗q′′0

(T ∗x′′

ε,δM ′′);

(2) c-Expq0C = z ∈ M |−∂q c(q0, z)∩C 6= ∅ ⊂ [∂cφ(Zε,δ,h)]x0 ⊂ ∂cφ(Zε,δ,h), where ∂q denotes

the subdifferential with respect to q variable;

(3) H n′′(C)H n′′

(π′′(Zε,δ,h)) & hn′′, where & is independent of h, δ, ε.

Proof. In the following, we first construct such a set C and then we show the desired properties.The set C will be given as a convex hull of certain covectors p1, . . . , pn′′ , see (6.22). We gothrough several steps.

First we find some auxiliary covectors p1, . . . , pn′′ . From Lemma 3.1 applied to the convexset π′′(Zε,δ,h), there is an ellipsoid E such that

E ⊂ π′′(Zε,δ,h) ⊂ n′′E (6.16)

where the scaling n′′E is with respect to the barycenter of the ellipsoid. Let p′′i , 1 ≤ i ≤ n′′,denote the unit orthogonal covectors parallel to the axes of the ellipsoid E, and denote by ai thelength of the i-th principal axis of E. Find hyperplanes Π′′

i ⊂ T ∗x′′

ε,δM ′′ that are orthogonal to p′′i

and touch tangentially the boundary of π′′(Zε,δ,h) at points q′′i , 1 ≤ i ≤ n′′. Let q′′0 be the pointin T ∗

x′′ε,δ

M ′′ corresponding to x0, and denote by `i the distance from q′′0 to Π′′i . Then, thanks to

(6.16) we have

n′′∏i

`i ≤n′′∏i

(2n′′ai) . H n′′(π′′(Zε,δ,h)). (6.17)

For each q′′i , there exists q′i ∈ T ∗x′

ε,δM ′ such that the hyperplane Πi := T ∗

x′ε,δ

M ′ × Π′′i ⊂ T ∗

xε,δM

tangentially touches the boundary ∂Zε,δ,h at the point qi = (q′i, q′′i ). Let xi = c-Expxε,δ

qi. Sincepi = (0, p′′i ) is orthogonal to Πi and Zε,δ,h is a sublevel set of the convex function ϕ, there existsa scalar multiple ti ∈ R+ such that tipi ∈ ∂ϕ(qi). By Assumption 2.2 and Loeper’s maximumprinciple (DASM) (Lemma 2.7), the point zi = c-Expqi

tipi satisfies zi ∈ ∂ cϕ(qi) = ∂cφ(xi).Note that in fact,

zi = c-Expqitipi = c-Expxi

η(tipi)

where η is the affine map given by Lemma 3.4 (in whose statement we replace x0, q0 and y0

with xi, qi and xε,δ, respectively). Moreover, using the decomposition

pi = (0, p′′i ) ∈ T ∗q′i

(T ∗x′

ε,δM ′) × T ∗

q′′i(T ∗

x′′ε,δ

M ′′),

27

we see that the c-segment (with respect to qi)

[0, 1] 3 t 7→ zi(t) = c-Expqi(1 − t)tipi = c-Expxi

((1 − t)η(tipi)

)from zi(0) = zi to zi(1) = xε,δ, is of the form

zi(t) = (x′ε,δ, z

′′i (t)) ∈ M ′ × M ′′.

Observe that by Proposition 6.3 and Assumption 2.3, we have

z′′i (t) ∈ M ′′(x′′), ∀t ∈ [0, 1],∀x ∈ Zε,δ,h. (6.18)

We use these c-segments zi(t) to define the points pi, i = 1, · · · , n′′. Define the function

mzi(t)(q) := −c(q, zi(t)) + c(qi, zi(t)) + ϕ(qi).

Clearly, mzi(0) ≤ ϕ and mzi(1) ≡ ϕ(qi) = h + ϕ(q0). By continuity there exists τi ∈ [0, 1) suchthat

mzi(τi)(q0) = ϕ(q0).

Also, Loeper’s maximum principle (DASM) implies

mzi( τi )

mzi(1)

mzi(0)

q0

pi

ϕ

qi

_

_

_

Figure 4: The supporting function mzi(0) = mzi touches ϕ at qi from below. By interpolating betweenmzi = mzi(0) and mxε,δ

= mzi(1) along the c-segment with respect to qi, we can find τi ∈ [0, 1) such thatmzi(τi)(q0) = ϕ(q0). Then the covector pi used to construct C is defined as pi := (0,−Dq′′ c′′(q′′0 , z′′i (τi)) ∈∂mzi(τi)(q0).

mzi(τi) ≤ max[h + ϕ(q0), ϕ],

so that in particular

mzi(τi) ≤ ϕ on ∂Zε,δ,h,

hence, by the definition of [∂cφ(Zε,δ,h)]x0 ,

zi(τi) ∈ [∂ cϕ(Zε,δ,h)]q0 = [∂cφ(Zε,δ,h)]x0 for every i = 1, . . . , n′′. (6.19)

For later use, consider the nonzero vectors

pi(τi) = (1 − τi)tipi

= (0, (1 − τi)tip′′i )=

(0,−Dq′′ c

′′(q′′i , z′′i (τi)))∈ T ∗

q′i(T ∗

x′ε,δ

M ′) × T ∗q′′i

(T ∗x′′

ε,δM ′′), i = 1, · · · , n′′. (6.20)

28

Clearly these vectors are all mutually orthogonal. Moreover, because

pi(τi) ∈ ∂mzi(τi)(qi), i = 1, · · · , n′′,

we have by the convexity of mzi(τi),

|pi(τi)| ≥ϕ(qi) − ϕ(q0)dist(q0, Πi)

=h

`i. (6.21)

To finish the construction of C, let

pi := (0,−Dq′′ c′′(q′′0 , z′′i (τi)) (6.22)

∈ T ∗q′0

(T ∗x′

ε,δM ′) × T ∗

q′′0(T ∗

x′′ε,δ

M ′′), i = 1, . . . , n′′.

Notice that zi(τi) = (x′ε,δ, zi(τi)′′) = c-Expq0

pi. Let C = co(p1, . . . , pn′′) be the convex hull ofp1, . . . , pn′′ . In the following, we will see that C satisfies the desired properties (1), (2) and(3). First, (1) follows immediately from (6.22), while (2) is a direct consequence of (6.19) andLemma 2.8.

Now, let us show (3). By (6.18) each zi(τi)′′ stays uniformly away (for small δ, h) from thecut-locus of π′′(Zε,δ,h). Hence we can apply Lemma 3.5 to (6.20) and (6.22) to see that pi is closeto pi(τi) when we use the canonical identification T ∗

qi(T ∗

xε,δM) ≈ T ∗

q0(T ∗

xε,δM); more precisely,∣∣pi − pi(τi)

∣∣ ≤ oh(1)|pi|,

where oh(1) is a quantity which goes to 0 as h → 0. Since the vectorspi(τi)

i=1,...,n′′ are all

mutually orthogonal, pi are almost mutually orthogonal covectors, which by (6.21) satify

|pi| = |p′′i | &∣∣pi(τi)

∣∣ ≥ h

`i.

(Here, for sufficiently small δ and hδ , the inequality & and the almost orthogonality are indepen-

dent of δ, h and ε.) This gives

H n′′(C) &

n′′∏i=1

h

`i.

This estimate combined with (6.17) shows (3). This completes the proof.

6.6 Proof of Theorem 6.4 (Alexandrov upper bound near cut-exposed point):final argument

In this section we finish the proof of Theorem 6.4. Let 0 < ε < 1, and fix 0 < δ ≤ δ1(ε) :=minδ0(ε), δ2(ε) and 0 < h ≤ h1(ε, δ) := minδ2, h2(ε, δ), with δ0(ε) and δ2(ε), h2(ε, δ) as inProposition 6.3 and 6.7 respectively. The estimates ., &, ≈ in this section are all independentof ε, δ and h.

To make use of the results of previous sections, we need the following comparison result:

Proposition 6.9. The set C constructed in Proposition 6.8 satisfies

H n′′(C) . |∂cφ(Zε,δ,h)|.

29

Note that even with Proposition 6.8 (2), this estimate is not obvious because n′′ < dimM .

Proof. For each ε, δ, h as in Proposition 6.8, we will find an auxiliary set A = Aε,δ,h ⊂ D in afixed (thus independent of ε, δ, h) compact set D ⊂ M∗(x0) ⊂ T ∗

x0M such that

c-Expx0(A) ⊂ [∂cφ(Zε,δ,h)]x0 ⊂ ∂cφ(Zε,δ,h); (6.23)

|A| & H n′′(C). (6.24)

Once such a set is constructed, the desired estimate follows from

|∂cφ(Zε,δ,h)| ≥ |c-Expx0(A)| & |A| (since A ⊂ D).

The construction of A goes through several steps. First, apply to the set C the (extended) mapp ∈ T ∗

q0(T ∗

xε,δM) 7→ η(p) ∈ T ∗

x0M as in Lemma 3.4 (with y0 = xε,δ), and let η(C) ⊂ T ∗

x0M denote

its image. Notice that by Proposition 6.8(2)

c-Expx0(η(C)) = c-Expq0

C ⊂ [∂cφ(Zε,δ,h)]x0 .

Let us compare H n′′(η(C)) with H n′′

(C). For each p = (0, p′′) ∈ C, Lemma 3.4 applies as

η(p) =(η′ε,δ , p′′

(−Dx′′Dx′′c′′(x′′

0, x′′ε,δ)

)+ η′′ε,δ

),

where ηε,δ = −Dxc(x0, xε,δ) (thus, c-Expx0(ηε,δ) = xε,δ). Therefore

η(C) ⊂ η′ε,δ × T ∗x′′0M

and

H n′′(η(C)) = |det Dx′′Dx′′c′′(x′′

0, x′′ε,δ)|H n′′

(C) ≈ H n′′(C).

Notice that x′′ε,δ is independent of ε (see (6.7)) and stays uniformly away from Cut(x′′

0), so thatthe above estimate is independent of ε, δ and h.

We now use a convexity argument to construct A. We will first construct some suitable setsC1, . . . , Cb0 , and C0, inside a fixed compact set (independent of ε, δ, h) in M∗(x0), which satisfythe properties of the sets Si in Lemma 3.3. These sets will also satisfy:

c-Expx0(C1) ∪ . . . ∪ c-Expx0

(Cb0) ∪ c-Expx0(C0) ⊂ [∂cφ(Zε,δ,h)]x0 ;

H ni(Ci) & 1, i = 1, . . . b0;

H n′′(C0) & H n′′

(η(C)).

Then A will be given as the convex hull of these sets, that is A = co(C1, . . . , Cb0 , C0). Byconvexity of M∗(x0), A will be in a fixed compact set, say D, independent of ε, δ, h, and thec-convexity of [∂cφ(Zε,δ,h)]x0 (see Lemma 2.8) will imply c-Expx0

(A) ⊂ [∂cφ(Zε,δ,h)]x0 , showing(6.23). We will then apply Lemma 3.3 to get

|A| & H n′′(η(C)) ≈ H n′′

(C),

which gives (6.24). Hence we are let to construct C1, . . . , Cb0 , C0.To construct C1, . . . , Cb0 , recall that M ′ = M1×. . .×M b0 , and for every i ∈ 1, . . . , b0 there

exists yi ∈ ∂cφ(x0) with yii = −xi

0. Moreover M iyi⊂ ∂cφ(x0) by Lemma 6.6. We further observe

30

that the same inclusion holds for all the components yli of yi that satisfy yl

i = −xl0. Hence, once

yi has a cut-locus component with x0, then one can change such component arbitrarily, andthe resulting point still remains inside ∂cφ(x0). Combining this fact with Loeper’s maximumprinciple (DASM) we can find a covector vi and a set Ci ⊂ ∂φ(x0) ⊂ T ∗

x0M , with vi ∈ Ci whose

components are either vli = 0 or vl

i ∈ M l∗(xl0), and

Ci = q ∈ T ∗x0

M | 2qi ∈ M i∗(xi0), and ql = vl

i for l 6= i .

Clearly, Ci is compact and Ci ⊂ M∗(x0). Moreover c-Expx0Ci ⊂ ∂cφ(x0) ⊂ [∂cφ(x0)]x0 and

H ni(Ci) & 1. Also, observe that the construction of C1, . . . , Cb0 is independent of ε, δ, h.Let us now construct the set C0. From Propositions 6.2 and 6.3 we see that for δ and

hδ sufficiently small there exists a compact set C ′′ ⊂ M ′′(x′′

0) (independent of ε, δ, h) withπ′′(∂cφ(Zε,δ,h)) ⊂ C ′′. Recall the definition of a0, b0, xε,δ = (x·

ε, x··δ ), and that x··

δ ∈ M ··(x··0).

Then we write ηε,δ = (η·ε, η··δ ) ∈ T ∗

x·0M · × T ∗

x··0M ·· and we observe that η··δ is uniformly away from

the boundary of M ··∗(x··0). These facts imply that there exists a compact set C ··

2 ⊂ M ··∗(x··0)

(independent of ε, δ, h) such thatη(C) ⊂ η·ε × C ··

2 .

However, η·ε → ∂M ·∗(x·0) as ε → 0, thus η(C) is not kept in a fixed compact set in M∗(x0).

In particular, we cannot take η(C) for C0, and this motivates the following: Since x·ε = −x·

0

and xε,δ ∈ [∂cφ(Zε,δ,h)]x0 , applying Lemma 6.6 as in the previous paragraph we see that the setM · × x··

δ, in particular, (x·0, x

··δ ) belongs to [∂cφ(Zε,δ,h)]x0 . This point (x·

0, x··δ ) corresponds to

the covector (0, η··δ ). Consider the cone co((0, η··δ ) , η(C)

), and define C0 as

C0 := co((0, η··δ ) , η(C)

)∩

(η·ε2

, q··)∈ T ∗

x0M | q·· ∈ T ∗

x··0M ··

.

By a simple geometric argument

H n′′(C0) & H n′′

(η(C)),

and moreover, since η·ε2 ∈ 1

2M ·∗(x·0), the set C0 is contained in a fixed compact set in M∗(x0)

independently of ε, δ, h. By c-convexity of [∂cφ(Zε,δ,h)]x0 ,

c-Expx0(C0) ⊂ [∂cφ(Zε,δ,h)]x0 .

Note that by construction this set C0, together with C1, . . . , Cb0 , satisfy the property of the setsSi in Lemma 3.3. Furthermore they are in a fixed compact set in M∗(x0) independent of ε, δ, h.This completes the proof.

Combining Propositions 6.9 and 6.8(3) we obtain

hn′′ . H n′′(π′′(Zε,δ,h))|∂cφ(Zε,δ,h)|. (6.25)

We will finish the proof by applying Proposition 6.7. First, we need some preliminary steps.Use the notation given in Section 6.5. Let Z ′

ε,δ,h be the slice of Zε,δ,h in M ′ × x′′0, that is

Z ′ε,δ,h := x′ ∈ M ′ |(x′, x′′

0) ∈ Zε,δ,h.

Then Z ′ε,δ,h is embedded via x′ 7→ −Dx′c′(x′, x′

ε.δ) into Z ′ε,δ,h ⊂ M ′∗(x′

ε,δ), where

Z ′ε,δ,h := q′ ∈ M ′∗(x′

ε,δ) | (q′, q′′0) ∈ Zε,δ,h.

Embed in the same way each Ziε,δ,h (see (6.13)), i = 1, . . . b0, into Zi

ε,δ,h ⊂ M i∗(xiε,δ).

31

Proposition 6.10. Assume that 0 < δ ≤ δ0 and 0 < h ≤ δ2, with δ0 as in Proposition 6.3.Then the following inequalities hold:(

minx′∈Z′

ε,δ,h

|det(Dx′Dx′c′(x′, x′ε,δ))|

)|Z ′

ε,δ,h| ≤ |Z ′ε,δ,h| ≤

(max

x′∈Z′ε,δ,h

|det(Dx′Dx′c′(x′, x′ε,δ))|

)|Z ′

ε,δ,h|,(min

xi∈Ziε,δ,h

|det(DxiDxici(xi, xiε,δ))|

)|Zi

ε,δ,h| ≤ |Ziε,δ,h| ≤

(max

xi∈Ziε,δ,h

|det(DxiDxici(xi, xiε,δ))|

)|Zi

ε,δ,h|,

where | · | denotes the Riemannian volume (in the appropriate submanifold).

Proof. From (6.14)Dx′q′ = −Dx′Dx′c′(x′(q′), x′

ε,δ),

and so the first inequality follows from

|Z ′ε,δ,h| =

∫Z′

ε,δ,h

|det Dx′q′| dx′.

The proof of the second inequality is analogous.

By convexity and Lemma 3.3 one has

H n′(Z ′

ε,δ,h) &b0∏

i=1

H ni(Ziε,δ,h),

while Propositions 6.10 and 6.7 implyb0∏

i=1

H ni(Ziε,δ,h) ≥

b0∏i=1

(min

xi∈Ziε,δ,h

|det(DxiDxici(xi, xiε,δ))|

)|Zi

ε,δ,h|

&[ b0∏

i=1

(min

xi∈Ziε,δ,h

|det(DxiDxici(xi, xiε,δ))|

)]hn′

εa0

Combining these estimates with (6.25) we get

hn′+n′′ . εa0

[ b0∏i=1

(min

xi∈Ziε,δ,h

|det(DxiDxici(xi, xiε,δ))|

)]−1

H n′(Z ′

ε,δ,h) H n′′(π′′(Zε,δ,h))|∂cφ(Zε,δ,h)|

. εa0

[ b0∏i=1

(min

xi∈Ziε,δ,h

|det(DxiDxici(xi, xiε,δ))|

)]−1

|Zε,δ,h||∂cφ(Zε,δ,h)| (by Lemma 3.2)

. εa0 |Zε,δ,h||∂cφ(Zε,δ,h)|,

where the last inequality follows from

|Zε,δ,h| .(

maxx∈Zε,δ,h

det(DxDxc(x, xε,δ)))|Zε,δ,h|

(see Proposition 6.10) and

maxx′∈Z′ε,δ,h

det(Dx′Dx′c′(x′, x′ε,δ))[ ∏b0

i=1

(minxi∈Zi

ε,δ,h|det(DxiDxici(xi, xi

ε,δ)|)] . 1 as δ, h

δ → 0

(see Propositions 6.2 and 6.3). This concludes the proof of Theorem 6.4, and Theorem 5.1 isproved.

32

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